Questions tagged [circulant-matrices]

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Block circulant matrix with some non-zero minors

Consider the following block circulant matrix over the field $\mathbb{F}_2$ \begin{equation*}M:= \begin{pmatrix} B_0 & B_1 & B_2 & B_3 & B_4 \\ B_4 & B_0 & B_1 ...
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Large submatrices of circulant matrices

What properties of circulant matrices are inherited by their principal submatrices? To be more specific: Given a square (zero-one) matrix $M$ of order $n$ with all elements on its main diagonal ...
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Fullrankness of sum of time shifts

I am working with finite Gabor frames and in this context a problem appeared which I am trying to solve for a couple of weeks now. Given a $(p,k,1)$ cyclic difference set for $\mathbb{Z}_p$ which is ...
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Lovasz theta and circulant graphs

Let $\theta(G)$ denote Shannon zero error capacity of graph $G$ and $\vartheta(G)$ be Lovasz upper bound for $\theta(G)$. Let $C_{2n+1}$ denote cycle graph with $2n+1$ nodes. We know following two ...
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Bounds for maximum determinant of circulant matrices

The Hadamard circulant conjecture states that there do not exist circulant Hadamard matrices with more than $4$ columns. An $n$ by $n$ Hadamard matrix where the entries are chosen from $\{-1,1\}$ ...
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About the properities of sum of powers of items in a polynomial

Given a polynomial $f_1=a_1x+a_2x^2+\cdots+a_{p-1}x^{p-1}\in\mathbb{Z}[x]/(x^{p}-1)$, with prime $p$, we may generate the other $p-1$ polynomials: \begin{eqnarray*} f_2&=&a_1^2x^2+\cdots+a_{p-...
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Are certain normal matrices circulant? (Part 2)

Let $\mathcal{F}$ denote the family of real normal matrices $A$ such that $A^TA=\begin{pmatrix} a & b \\ b & \ddots \end{pmatrix}$, for $b>0$. As a user observed in the solution of Part 1 ...
Let $A=\{a_{ij}\}$ be a normal matrix such that $a_{ij}\geq 0$ with equality iff $i=j$. Suppose that $$A^TA=\begin{pmatrix} a & b & \cdots & b\\ b & a & \ddots & \vdots\\ \... 2answers 2k views How to determine if there exists a non-zero vector in the kernel If you are given a 0-1 circulant matrix with n rows and n columns, is there an efficient way of determining if there exists a non-zero \{-1,0,1\}-vector in its kernel? Could this problem ... 1answer 250 views Graphs with circulant distance matrices The cycle has this property. For instance, the distance matrix for a 6-cycle is: A=\begin{bmatrix} 0 & 1 & 2 & 3 & 2 & 1 \\\\ 1 & 0 & 1 &... 0answers 176 views bounds on the entries of an inverse circulant matrix Suppose that C is a (real) circulant invertible matrix defined by a vector d. Then C^{-1} is also a circulant defined by some vector f. There exists a standard formula that expresses the ... 4answers 6k views Eigenvectors and eigenvalues of a tridiagonal Toeplitz matrix Is it possible to analytically evaluate the eigenvectors and eigenvalues of the following n \times n tridiagonal matrix$$ \mathcal{T}^{a}_n(p,q) = \begin{pmatrix} 0 & q & 0 & 0 &...
Let $G$ be a cirulant graph with no loops at vertices and vertex degree $d$. Is the Lovasz theta function of this graph given by: $\vartheta(G) = \max_{i}\frac{-N\epsilon_{i}}{-\epsilon_{i}+d-1}$? ...